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International Journal of Pure and Applied Mathematics
Volume 89 No. 1 2013, 55-70
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: http://dx.doi.org/10.12732/ijpam.v89i1.7
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CONSECUTIVE EVEN NUMBER FINDING GRAPH
(CENFG) RELATED TO GOLD BACH CONJECTURE
Rongdeep Pathak1 § , Bichitra Kalita2
1,2 Department
of Computer Application (M.C.A)
Engineering College
Guwahati, 781013, Assam, INDIA
1,2 Assam
Abstract: In this paper, the consecutive even number finding graph has been
discussed. The solution of Goldbach conjecture has been established from this
graph.
AMS Subject Classification: 11A41, 11A51, 11A051, 05C30, 05C45
Key Words: the graph CPVEEWGS, CENFG, adjacency matrix
1. Introduction
It has been found that Pure Mathematics is mainly associated with abstract
concepts. From the eighteenth century onwards, this was a recognized branch of
combinatorial activities in various fields such as astronomy, physics, engineering
and so on. Finance and cryptography are current example of areas to which
pure mathematics is applied in significant ways.
In the branch of Pure Mathematics, the Number theory, which is again
playing a very important role in other branches of science and technology occupying a field of conjectural problems. Since the discovery of Number theory
it has been occupying a field of so many conjectural problems, which are yet
to be proved. There are many questions around Prime numbers, which are remained open, such as Gold Bach’s conjecture [1742], which proposes that every
even integer greater than 2 can be expressed as sum of two primes, Twin prime
Received:
June 18, 2013
§ Correspondence
author
c 2013 Academic Publications, Ltd.
url: www.acadpubl.eu
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R. Pathak, B. Kalita
conjecture which says that there are infinitely many pairs of primes whose difference is 2 and there always a prime between n2 and (n+1)2 and so on. Many
attempts have been made for the proof of these conjectural problems. But,
no acceptable solution has been obtained till now or nobody can find a suitable method to carry on this research depending on the available theorems and
results.
Comparing the graph theory with number theory, it has been found that
the graph theory, which is assumed to be applied, has started before number
theory. We know that graph theory started in 1736 from the Euler works on
Konigsberg bridge problem. But, interesting relation between graph theory and
number theory has not been found except the graphical partition. Hence there
is a field of study for existence any important theoretical results between them.
Graph theoretical ideas are highly utilized by computer science application such
as data mining, database, networking and so on. The concept of graph coloring
is utilized in resource allocation and scheduling. Path, walk and circuits in
graph theory are used in application of travelling salesman problem, database
design concept, networking etc. Here we combine the concept of graph theory
and number theory, a branch of pure mathematics for proving the conjectural
problem of Gold Bach conjecture. Kaida Shi [13] forwarded a new method to
prove Goldbach conjecture and twin prime conjecture. A new direction of proof
of Goldbach conjecture has been forwarded by Kalita [4] in 2000. Kalita [5] has
also been forwarded the proof of Goldbach conjecture in opposite direction.
After the publication of proof of Goldbach conjecture forwarded by Kalita,
Alex Vand etal [6] showed that the Goldbach conjecture is true for all the even
numbers from 2 to 1018 . Recently Kalita [12] has been forwarded the proof of
Goldbach conjecture i.e. every even number n greater than 2 can be expressed
as a sum of two primes, with the help of graph theoretic concept. Kalita etal
has been discussed some properties of even numbers with various properties of
graph and an algorithm has also been forwarded to know the various structure
of graphs relating to even numbers 2n + 4, 4n + 4, ( n + 1) ( n+ 2) and 6n +
2 for n≥1.
In this paper, the proof of Goldbach conjecture, which proposes that every
even integer greater than 2 can be expressed as sum of two primes, has been forwarded with the help of consecutive even number finding graph (CENFG). Two
theorems has been proposed and proved which actually give the new direction
of proof of Gold Bach Conjecture.
CONSECUTIVE EVEN NUMBER FINDING GRAPH...
57
Figure 1
2. Definitions, Theorems and Algorithm
Before going to forward the proof of Goldbach conjecture with the help of
consecutive even number finding graph, which proposes that every even integer
greater than 2 can be expressed as sum of two primes the following definitions
and terminologies have been cited to understand the use of graph for Gold Bach
conjecture.
Definition 1. A graph G = (V, E) consists of a set of objects V= {V1 , V2 ,
V3 . . . }called vertices, and another set E = {e1 , e2 , e3 ,. . . .}, whose elements are
called edges, such that each edge ek is identified with an unordered pair (vi , vj )
of vertices. The vertices vi , vj associated with edge ek are called end vertices
of ek . Graph whose edges or vertices have names are known as labeled. Graph
labeling usually refers to the assignment of labels to the edges and vertices of
a graph, subject to certain rules depending on the situation.
Definition 2. A graph G (V, E), in which there exists an edge between
every pair of vertices is called a complete graph.
Definition 3. An edge having the same vertex as both its end vertices is
called a self loop.
The graph of five vertices and seven edges with one self loop is shown in
Figure 1.
We now remind the following definitions [12] for clarification.
Definition 4. (Prime Vertex and Even Edge Weighted Graph, see [12]) Let
G (V, E) be a graph having at least one self loop without parallel edges. For the
graph G (V, E) we consider the set of vertices V = {V1 , V2 , V3 ,. . . . Vn }and
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R. Pathak, B. Kalita
Figure 2
the set of edges E = {E1 , E2 , E3 ,. . . .. En }. Let P1 <P2 <P3 <. . . . <Pn
for n≥5 be consecutive primes ≥ 13. Now attaching these primes P1 <P2 <P3
<. . . . <Pn for n≥5 with the vertices V1 , V2 , V3 ,. . . . Vn of the graph G which
are now called prime weight(PW) of the vertices and considering weights W1 ,
W2 , W3 , . . . .., Wn for the edges E1 , E2 , E3 ,. . . ..En , where the weights W1 ,
W2 , W3 , . . . .., Wn are obtained as a sum of any two prime weights Pi + Pj
of attached vertices and considering self loop, if exist in the graph as sum of
the same prime weight of the attached vertex, we can construct a new graph
with prime vertex set VatP and the weighted edges set EsoP and this graph thus
obtained is called a prime vertex and even edge weighted graph of the graph
G ( V,E ) and it is denoted by the graph PVEEWG (VatP, EsoP ). Figure 2 is
a prime vertex and even edge weighted graph (VatP, EsoP )for the five primes
13<17 <19 <23 <29. For this graph VatP = {13, 17, 19, 23, 29 }and EsoP =
{W1 = 30, W2 = 34, W3 = 36, W4 = 40, W5 = 42, W6 = 42,
W7 = 52 }and the original graph (see Figure 2) has one self loop and this
gives an edge of weight W2 = 34.
Definition 5. (Complete Prime Vertex and Even Edge Weighted Graph,
see [12]) A graph PVEEWG (VatP, EsoP ) is called a complete prime vertex and
even edge weighted graph if it is obtained from a complete graph G ( V, E ).
It is denoted by the graph CPVEEWG (VatP, EsoP ). Figure 3 is a complete
prime vertex and even edge weighted graph for the five primes 13<17 <19 <23
<29.
Definition 6. (Complete Prime Vertex and Even Edge Weighted Graph
with Self Loop) Let G (V, E) be a Complete Prime Vertex and Even Edge
Weighted graph [12]. We now introduce self loops to all vertices of the graph
CONSECUTIVE EVEN NUMBER FINDING GRAPH...
59
Figure 3
and attached all the consecutive primes to all vertices, where the self loop
means addition of same prime for attached weight (Prime) and the edges are
represented by the sum of two primes and the graph thus obtained is called
Complete Prime Vertex Even Edge Weighted graph with Self loop which is
denoted by the graph CPVEEWGS (V, E).
Figure 4 is a complete prime vertex and even edge weighted graph with self
loop for the five primes 13<17 <19 <23 <29.
It is observed that the graph CPVEEWGS (V, E) of Figure 4 always gives
even numbers 26, 30, 32, 36, 46, 42, 48, 52, 40, 42, 46, 58, 36, 38, 34 for
n=5 consecutive primes, which represents the edges of the CPVEEWGS graph
which are actually obtain as a sum of two prime, where V is the set of vertices
13, 17, 19, 23, 29. But, here, we found that all the even numbers are not
consecutive starting from 26 to 58. Interestingly the structure of the graph
(CPVEEWGS) is found as a graph CPVEEWGS(X, (X2 +X)/2), where X= 5+
m*n for m=n=0, 1, 2, 3. . .
This graph helps us to find out the consecutive even numbers (even weighted
edge) up to a certain limit, i.e. the graph of Figure 4 give consecutive even numbers 30, 32, 34, 36, 38, 40, 42. Here we do not consider the even number 26 for
counting the consecutive even numbers as we shall start our investigation from
the even number 30. When we consider the value of n=6 consecutive primes, we
have the graph CPVEEWGS(X, (X2 +X)/2) from where the consecutive even
numbers (even weighted edges) 30, 32, 34, 36, ..., 62 are to be counted but
not starting even number 26. This graph also gives consecutive even numbers
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Figure 4
starting from 32 to 54, a total 17 consecutive even numbers.
Observing this formulation of even numbers we have the following important
theorem.
Theorem 1. Consecutive even numbers can be calculated out of all even
numbers obtained from the graph G (X, (X2 +X)/2), where X= 5+m*n for
m=n=0, 1, 2, 3. . . up to some certain limit.
Proof. We prove the theorem by induction methods. It is found that the
result is true for X=5, that is for m=n=0. That is when we construct the graph
CPVEEWGS (X, (X2 +X)/2), for X = 5, we have the structure of the graph as
shown in Figure 4 from where fifteen even numbers 26, 30, 32, 36, 46, 42, 48,
52, 40, 42, 46, 58, 36, 38, 34 are obtained. Out of these fifteen even numbers
we can find seven consecutive even numbers 30, 32, 34, 36, 38, 40, 42 (omitting
only 26 as discussed previously). It is observed that for X = 6, that is for the
values of m=n=1, we have thirteen consecutive even numbers 30, 32, 34, 36, 38,
40, 42, 44, 46, 48, 50, 52, 54 from the graph CPVEEWGS (6, 21). (Note that
all the even numbers are nothing but the edges of the graph CPVEEWGS).
Let the theorem be true for X=5 +k*k for the values of m=n=k. That
CONSECUTIVE EVEN NUMBER FINDING GRAPH...
61
Figure 5
is for the graph CPVEEWGS (5 + k*k, ((5+k*k)2 +(5+k*k))/2), some consecutive even numbers can also be obtained. Now we shall prove it for X=
5 + (k+1)*(k+1) for the value of m=n=k+1. Here the term {(5 + (k+1)2
}2 + 5+ (k+1)2 )/2 which are the number of even numbers/edges of the graph
CPVEEWGS( 5 + (k+1)2 , (5 + (k+1)2 )2 + 5+ (k+1)2 )/2 ). It is observed
that the result is true for k=0, that is the graph CPVEEWGS has six number
of vertices when m=n=1 and hence our result is true for m=n=k=1.According
to the definition of the graph CPVEEWGS, the edges represents the even numbers, not all consecutive and from the even numbers and we can calculated some
consecutive even numbers. The following graph CPVEEWGS (X, (X2 +X)/2)
for infinite values of X can be considered for the existence of some consecutive
even numbers, where X= {P1, P2, P3 ,. . . . . . . . . . . . }and (X2 +X)/2 = {P1 +P2 ,
P1 +P3 ,. . . . . . . with P1 +P1 , P2 +P2 ,. . . ., Pn +Pn ,. . . . . . . . . }are the edges of the
complete graph with self loop.
The graph CPVEEWGS (X, (X2 +X)/2), gives some even numbers and out
of these some even number, we can calculate the consecutive even numbers up
to a certain limit.
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Hence the theorem i sproved.
The above theorem gives the information of even numbers/edges from where
certain consecutive even numbers can be calculated which are nothing but sum
of two primes and hence we require some other method from where one can
find out the all consecutive even numbers/edges. Here, we use one important
algorithm to get one adjacent matrix from where another structure of graph
called consecutive even number finding graph (CENFG), can be constructed
which will be discussed later.
Let us first develop the following algorithm.
Algorithm 1. Construction of adjacent matrix for even number Finding
graph (CENFG)
INPUT: Number of prime, N
OUTPUT: Find the adjacency matrix with self loop for CENFG
Step 1. Store N consecutive prime numbers (P1 <P2 <P3 <. . . <PN ) in an
array A[] of size N, where P1 =13 and N≥5.
Step 2. Calculate the sum of two prime numbers and result store in a 2 D
array B[][] of size N*N.
1. Repeat for I = 1 to N
2. Begin
3. Repeat for J = 1 to N
4. Begin
5. B[I][J] = A[I] + A[J]
6. End
7. End
Step 3. Store the values of the 2 D array B[][] in to an another array C[] of
size N*N
1. Repeat for I =1 to N
2. Begin
3. Repeat for J =1 to N
4. Begin
CONSECUTIVE EVEN NUMBER FINDING GRAPH...
5. C[K] = B[I][J]
6. K= K + 1
(Initial value of K is 0)
g. End
h. End
Step 4. Sorting the value of C[N*N] in an ascending order.
1. Repeat for I =1 to (N*N – 1)
2. Begin
3. Repeat for J =(I + 1) to N*N
4. Begin
5. If ( C[I] >C[J])
6. Temp1 = C[I]
7. C[I] = C[J]
8. C[J] = Temp1
9. End
10. End
Step 5. Remove duplicate values from C[N*N].
1. M= N*N
2. Repeat for I = 1 to M – 1
3. Begin
4. Repeat for J = I + 1 to M
5. Begin
6. If ( C[I] = C[J])
7. M= M – 1
8. Repeat for K = J to M
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R. Pathak, B. Kalita
9. Begin
10. C[K] = C[K + 1]
11. End
12. J = J – 1
13. End
14. End
Step 6. Except the first set of consecutive even numbers of the array C[N*N]
(not considering the first value i.e. 26) change the values of the remaining even
numbers to zero.
Step 7. Largest value of C[N*N] is stored in TEMP
Step 8. Change the values of B[][] to zero which are larger than TEMP
value.
1. Repeat for I = 1 to N
2. Begin
3. Repeat for J = 1 to N
4. Begin
5. If ( B[I][J] >TEMP )
6. B[I][J] = 0
7. End
8. End
Step 9. Construct an intermediate matrix from which one can finally develop an adjacency matrix.
1. Initialize B[1][1] = 0
2. Repeat for I = 1 to N
3. Begin
4. Repeat for J = 1 to N
CONSECUTIVE EVEN NUMBER FINDING GRAPH...
65
5. Begin
6. If B[I][J] == 0 go to step 9.d
7. Repeat for X = 0 to N
8. Begin
9. Repeat for Y = 0 to N
10. Begin
11. If (I == X && J == Y)
12. Go to step 9.i
13. Else
14. If ( B[I][J] == B[X][Y] && ( I+X) 6=(J+Y) )
15. B[X][Y] =0
16. End
17. End
18. End
19. End
Step10. Construct the final Matrix from which we can draw Consecutive Even
Number Finding (CENFG) graph.
1. Repeat for I = 1 to N
(a) Begin
i.
ii.
iii.
iv.
v.
(b) End
Repeat for J = 0 to N
Begin
If ( B[I][J] ≥ 1 )
B [ I][J ] = 1
End
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R. Pathak, B. Kalita
P1
P2
P3
P4
P5
P6
=
=
=
=
=
=
13
17
19
23
29
31
P1 = 13
0
1
1
1
1
1
P2 = 17
1
1
0
1
1
1
P3 =19
1
0
1
0
0
1
P4 =23
1
1
0
0
1
1
P5 =29
1
1
0
1
0
0
P6 =31
1
1
1
1
0
0
Table 1
Step 11. STOP
From the algorithm, we can find an adjacency matrix which is nothing
but a representation of vertices/primes and the interconnection between vertices/primes. This adjacency matrix gives the total number of edges of the
graph CENFG (V, E), which are consecutive even numbers which are actually
found as a sum of two primes. The number of edges (P), which is found from
the adjacency matrix always greater than the value of N, which is the input
to the algorithm. Here, N is the number of consecutive primes/vertices of the
graph CENFG (X, P), which can be drawn from the adjacency matrix.
Here let us take one example for N=6, say for six consecutive primes starting
from 13.
After applying the algorithm, we have the following table (1) which gives
the adjacency matrix with self loop obtained from the algorithm from which we
can draw the consecutive even number finding graph CENFG (V, E) as shown
in Figure 6.
The following graph of Figure 6 is a consecutive even number finding graph
(CENFG) which is drawn with the help of table 1 and from this graph one
can find out the consecutive even numbers which are obtained as a sum of two
primes/vertices.
The above graph CENFG (6, 13) gives thirteen consecutive even numbers
/edges which are obtained as a sum of two primes/vertices and this gives the
proof of Goldbach conjecture for thirteen consecutive even numbers which can
be expressed as a sum of two primes. Hence the following theorem can be
considered for counting all consecutive even numbers/edges.
We can now define the consecutive even number finding graph as follows:
Definition 7. (Consecutive Even Number Finding Graph) A prime vertex
and even edge weighted graph [12] G (V, E) is called a consecutive even number
CONSECUTIVE EVEN NUMBER FINDING GRAPH...
67
Figure 6
finding graph if all the edges of the graph gives consecutive even numbers and
this graph is denoted as a graph CENFG (V, E) where V is the number of
attached vertices and E is the number of edges/even number.
Figure 7 is a graph CENFG (21, 76). This graph gives seventy six consecutive even numbers/edges, starting from thirty to one hundred eighty, which are
obtained from the algorithm.
Theorem 2. The graph CENFG (X, P), where X=5 +m*n, m=n=0, 1, 2,
3. . . and P is the value that can be calculated from the matrix obtained from
the algorithm, always gives consecutive even numbers/edges which are the sum
of two primes.
Proof. It is found from the Algorithm 1 that P is the number of edges
which can be calculated from the adjacency matrix and P always represents
consecutive even numbers starting from thirty.
Now, if we put the value of X=6, that is m=n=1, we can find an adjacency
matrix as shown in table 1 and from this table, we can draw a graph CENFG
(6, 13) as shown in Figure 6. This graph gives thirteen even weighted edges/
numbers, which are 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52 and 54.
Continuing the process of construction of the graph CENFG for X ≥ 6
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R. Pathak, B. Kalita
Figure 7
and m=n≥2, we can have the consecutive even numbers /edges to any limit.
For example, if we put the value of X=21, that is m=n=4, we can find another
graph CENFG (21, 76) as shown in Figure 7. This graph CENFG (21, 76) gives
seventy six even weighted edges/ numbers starting from thirty to one hundred
eighty.
We see that, the number of even weighted edges/numbers gradually increasing when the value of X increases. Again it is found from the algorithm that
the value of P, which is the number of even weighted edges/numbers always
greater than the value of N, which is the number of vertices/primes.
Hence, the graph CENFG (X, P), always gives consecutive even numbers/edges, which are the sum of two primes/vertices.
CONSECUTIVE EVEN NUMBER FINDING GRAPH...
69
Gold Bach Conjecture. Every even number greater than 2 can be expressed as a sum of two primes.
Proof. From the theorem 2, we can find that the graph CENFG (X, P),
gives infinite consecutive even numbers/edges that is values of P for infinite
value of X. This immediately satisfies that every even number greater than 30
can be expressed as sum of two primes, which completes the proof of Gold Bach
Conjecture.
Note. Here we should not consider the smaller values of even number from
2 to 28 as they are easy to express as a sum of two primes.
3. Conclusion
In this paper, the proof of Gold bach conjecture, which proposes that every
even integer greater than 2 can be expressed as sum of two primes, has been
forwarded with the help of consecutive even number finding graph (CENFG).
Here, the solution of Gold bach conjecture is established with the help of graph
theory.
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